The condition for the emergence of a Turing instability can be assessed
by performing a linear stability analysis about the homogeneous equilibrium [1]}, [2]}, [3]}, [4]}, as previously shown. Assuming \(\mathbf {G}\)  to be the identity function and the reference orbit to coincide with a stable stationary equilibrium \(\mathbf {s}_0=(u_0,v_0)\) , Eq. (REF ) simplifies into:
\({\left\lbrace \begin{array}{ll}\dot{\delta u_i} &=\partial _u f(u_0,v_0)\delta u_i +\partial _v f(u_0,v_0)\delta v_i +D_u \sum _j L^H_{ij}\delta u_j\\\dot{\delta v_i} &=\partial _u g(u_0,v_0)\delta u_i +\partial _v g(u_0,v_0)\delta v_i +D_v \sum _j L^H_{ij}\delta v_j\, ,\end{array}\right.}\) 
